In NP formalism, Einstein's equation can be transformed into Teukolsky equations [1]}, [2]}, [3]}. The ten independent components of the Weyl tensor are converted into five complex Weyl scalars. Among them, we are interested in the Weyl scalar \(\psi _4\)  which is interpreted as the outgoing transverse radiation [4]}. We further define \(\psi =\rho ^{-4}\psi _4\)  with \(\rho = -\left(r-ia\cos \theta \right)^{-1}\) , which satisfies the Teukolsky equation,
\(\begin{aligned}&\left[\frac{\left(r^{2}+a^{2}\right)^{2}}{\Delta }-a^{2}\sin ^{2}\theta \right]\frac{\partial ^{2}\psi }{\partial t^{2}}+\frac{4Mar}{\Delta }\frac{\partial ^{2}\psi }{\partial t\partial \varphi }+\left[\frac{a^{2}}{\Delta }-\frac{1}{\sin ^{2}\theta }\right]\frac{\partial ^{2}\psi }{\partial \varphi ^{2}}-\Delta ^{2}\frac{\partial }{\partial r}\left(\Delta ^{-1}\frac{\partial \psi }{\partial r}\right)-\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial \psi }{\partial \theta }\right)\\&\hspace{56.9055pt}+4\left[\frac{a(r-M)}{\Delta }+\frac{i\cos \theta }{\sin ^{2}\theta }\right]\frac{\partial \psi }{\partial \varphi }+4\left[\frac{M\left(r^{2}-a^{2}\right)}{\Delta }-r-ia\cos \theta \right]\frac{\partial \psi }{\partial t}+\left(4\cot ^{2}\theta +2\right)\psi =4\pi \Sigma T,\end{aligned}\) 
